The equality introduction rule is reflexivity: any well-formed first-order term is equal to itself, with no premises. The equality elimination rule permits replacement of equals in a first-order formula:
(1)
In the second rule, write and as the results of substituting and into the same designated free occurrences of a variable in . Both substitutions must avoid variable capture. The rule preserves the surrounding assumptions. This expresses first-order equality rather than treating equality as an arbitrary relation symbol.

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